Physics 6B - 1st Midterm 2016 Dr. Nassar (1-hour exam -- 3x5 card allowed) NAME:_________________________ Complete the following 4 problems. 1) (15 points) A 1.15-kg mass oscillates according to the equation x(t) = 0.65 cos (7.4 t) where x is in meters and t in seconds. Determine a) (2.5 points) the amplitude, A = 0.650m b) (2.5 points) the frequency, f = (ω/2π) = (7.4/2π) = 1.18 Hz c) (5 points) the total energy (is the same at any point – the same at maximum potential energy) E = (1/2)kA2 = (1/2)mω2A2 = (1/2) (1.15)(7.4)2 (0.65)2 = 13.3 J d) (5 points) the potential energies when x = 0.26 m. U = (1/2)kx2 = = (1/2)mω2x2 = (1/2) (1.15)(7.4)2 (0.26)2 = 2.1 J 2) (15 points) Two waves traveling in opposite directions on a string are described by y1 (x,t) = (0.2 m) sin (2x – 4t) and y2 (x,t) = (0.2 m) sin (2x + 4t). They produce a standing wave. Determine a) the function for the standing wave, b) the maximum amplitude at x = 0.45 m and c) transverse velocity of a point in the string at at x = 0.45 m and t = π/8 s. a) (5 points) k = 2 m-1 and = 4 s-1

2 sin cos (0.4 )sin 2 cos 4y A kx t m x tω= =

b) (5 points) (0.4 )sin 2(0.45)cos 4 0.31cos 4y m t t= = Maximum amplitude is 0.31 m and occurs when cos(4t) = 1 or t = 0.

c) (5 points) (0.4 )(4)sin 2 sin 4

(0.4 )(4)sin 2(0.45)sin(4 / 8) 1.25m/ s

y

y

yv m x tt

v m π

∂= = − =

∂= − = −

ANSWERS

3) (10 points) A 5000-Hz sound wave is emitted by a stationary source. This sound wave reflects from an object moving 3.5 m/s toward the source. What is the frequency of the wave reflected by the moving object as detected at rest near the source?

343 3.5' 5000 5051

343" ' 5051 5103343 3.5

343 3.5" 5000 5103343 3.5

L

S

L

S

c vf f Hzc ccf f Hz

c vor

c vf f Hzc v

+ += = =

= = =− −

+ += = =

− −

4) (5 points) A 0.32-m-long violin string is tuned to play a note A (440 Hz). a) What is the wavelength of the fundamental mode of vibration and b) what is the velocity of the wave? a) (2.5 points) λ = 2L = 2(0.32) = 0.64 m b) (2.5 points) v = λ f = (0.64)(440) = 281.6 m/s Multiple-choice question: (5 points) An object is attached to a spring and oscillates with amplitude A and period T, as represented on the graph below. The nature of the velocity v and acceleration a of the object at time T/2 is best represented by which of the following?

(a) v > 0, a > 0 (b) v > 0, a < 0 (c) v < 0, a = 0 (d) v = 0, a < 0 (e) v = 0, a > 0